In physical systems that are assembled as a lossless interconnection of physical subsystems, the total power consumed or produced by the interconnection is zero, i.e., power is conserved. A lossless physical interconnection of K subsystems, each with conjugate effort and flow variables denoted ek and ƒk, respectively, therefore has a conservation law that may be written as the equation, e1ƒ1+ . . . +eK ƒK=0. In such physical systems, this equation holds independent of whether the interconnected subsystems are linear or nonlinear, time-invariant or time-varying, or deterministic or stochastic. As such, the use of the conservation equation in the derivation of useful mathematical theorems about physical systems often implies not only that the theorems apply very broadly, but also that the application of linear or nonlinear transformations may be used as a tool in the corresponding derivations.
One may look to electrical networks to find a broad class of such theorems originating from equations of the form of the conservation equation. In this class of physical systems, the equation is embodied by Tellegen's Theorem, and a comprehensive summary of many of the accompanying theorems, which address among other things stability, sensitivity, and variational principles in electrical networks, can be found in Tellegen's Theorem and Electrical Networks (Penfield, Spence and Duinker, The MIT Press, Cambridge, Mass., 1970).
In contrast to physical systems, many current signal processing architectures, including general-purpose computers and digital signal processors, implement algorithms in a way that is often far-removed from the physics underlying their implementation. One advantage to this is that a wide range of signal processing algorithms can be realized that might otherwise be difficult or impossible to implement directly in discrete physical devices, including for example transform-based coding, cepstral processing, and adaptive filtering. However, the high degree of generality facilitated by these types of architectures comes with the expense of losing some of the powerful analytic tools traditionally applied in the design and analysis of the restricted set of systems that is allowed physically, and derivations of many of these tools stem from equations of the form of the conservation equation.
A common strategy to overcome this essentially involves designing signal processing algorithms that mimic the equations or sets of equations describing a specific physical system or class of physical systems. Any signal processing algorithm that can be put in the form of the equations is then regarded as being of a special class, to which a wide range of theorems often apply. For example, the class of signal processing systems consisting of two subsystems interconnected to form a feedback loop is a canonical representation into which it is often desirable to place control systems, and about which many useful results are known. This strategy has also been used with respect to the wave-digital class of structures as well as network-based optimization algorithms. Indeed, conservation principles are at work in a wide class of useful systems and algorithms.